Capacity Distortion by Inner Functions in the Unit Ball of C~n

摘要

An inner function is a bounded holomorphic function from the unit ball B_n of C~n into the unit disk Δ of the complex plane such that the radial boundary values have modulus 1 almost everywhere. If E is a nonempty Borel subset of partial derivΔ, we denote by f~(-1)(E) the following subset of the unit sphere S_n of C~n: f~(-1)(E) = {ξ ∈ S_n : lim_(f→1) f(rξ) exists and belongs to E }. There is a classical lemma of Loewner (see e.g. [R, p. 405; T, p. 322]), about the distortion of boundary sets under inner functions.